Wallpaper groups

The diversity of wallpaper groups isn't exhausted with our observations so far. Wallpaper groups play a major role in condensed matter physics and are therefore often also called "crystallographic groups".

We want to briefly define what a wallpaper group is: A wallpaper group is a set

of geometric transformations, which first of all is a group in the mathematical sense, i.e. for every two transformations $a,b\in G$ we define the composition $a \circ b$ as operation. This operation has to fulfill the following three axioms:

There is an identity element $e\in G$ , i.e. $a\circ e=a$ holds for all $a\in G$
For every element $a\in G$ there exists an inverse element $a^{-1}$ , i.e. $a\circ a^{-1}=e$
The operation is associative, i.e. $a\circ (b\circ c)=(a\circ b)\circ c$ holds true

These properties are automatically fulfilled for geometric transformations, as long as the inverse transformation of every transformation is contained within the group as well, and furthermore the composition of any two transformations (which is again a transformation) is also contained in the group.

For wallpaper groups, the following additional conditions have to be fulfilled:

There have to be translations in different directions.
Only distance-preserving transformations are allowed (i.e. the congruence mappings: rotation, reflection, translation and glide reflection)
Mapping a single point of the plane using all elements of the group, a sufficiently small area around the original point has to be devoid of any image points except the original point itself.

The first condition ensures that the group will cover the whole plane. The second condition rules out contractions, expansions and circle inversions. The third condition ensures that the operations "add up" properly - points shouldn't get mixed up arbitrarily.

As there were only three different kinds of triangle kaleidoscopes, there is only a finite number of structurally different wallpaper groups: there are 17.

Out of these 17 groups we already know 5:

The group consisting only of two directions of translations,
the group with the rectangular mirror box and
the three kaleidoscope groups.

We don't want to introduce all the remaining 12 groups, but still give representations of a few particularly beautiful specimens. The following applet shows Dr. Stickler copied all over the plane by different wallpaper groups. One can change the group by pressing one of the buttons. Dr. Stickler can be moved around as usual. This time his left foot will move him as a whole.

Please enable Java for an interactive construction (with Cinderella).

A very creative interaction with wallpaper groups is possible using this Ornament program. The program was written by Martin von Gagern for the mathematical exhibition ix-quadrat at the TU München. In that program one can choose a wallpaper group and draw an ornament using that group. the resulting images have high aesthetic appeal.

Download file:

SymmStickler.cdy

Kaleidoscopes $\hookleftarrow$ Contents $\hookrightarrow$ Iterated similarities